Galaxy populations in galaxy clusters selected by the Sunyaev-Zeldovich Effect

Galaxy Populations in Galaxy

Clusters Selected by the

Sunyaev-Zeldovich Effect

Galaxy Populations in Galaxy

Clusters Selected by the

Sunyaev-Zeldovich Effect

Alfredo Andr´

es Zenteno Vivanco

Dissertation an der Fakult¨

at f¨

ur Physik

Dissertation at the Faculty of Physics

der Ludwig-Maximilians-Universit¨

at M¨

unchen

of the Ludwig Maximilian University of Munich

f¨

ur den Grad des

for the degree of

Doctor rerum naturalium

vorgelegt von Alfredo Andr´

es Zenteno Vivanco

presented by Alfredo Andr´es Zenteno Vivanco

aus Concepci´

on, Chile

from Concepci´on, Chile

1st Evaluator: Prof. Dr. Joseph Mohr 2nd Evaluator: Prof. Dr. Andreas Burkert Date of the oral exam: 24.02.2014

vii

Zusammenfassung

Wir pr¨asentieren eine Studie ¨uber die Galaxienpopulation in massereichen Galaxienhaufen, ausgew¨ahlt aufgrund ihrer Signatur im Sunyaev–Zel’dovich Effekt (SZE). Die Auswahl mit-tels des SZE ist ann¨ahernd massenlimitiert, d.h. das untere detektierbare Massenlimit variiert nur geringf¨ugig mit der Rotverschiebung, wodurch der SZE eine ideale Auswahlmethode f¨ur ein Studium der Entwicklung von Galaxien innerhalb einer Haufenumgebung ist. Wir be-ginnen diese Arbeit mit einer Einf¨uhrung in den SZE, einer Vorstellung des S¨udpolteleskops (SPT) mittels dessen der SZE gemessen werden kann, sowie des Forschungsprojekts, inner-halb dessen diese Arbeit entstanden ist. Im Folgenden pr¨asentieren wir dann die Studien zur Galaxienpopulation, die der Kern dieser Doktorarbeit sind.

In Kapitel 3 pr¨asentieren wir die erste großskalige Folgestudie eines mittels des SZE aus-gew¨ahlten Galaxienhaufen–Samples. Von 224 Galaxienhaufen–Kandidaten des Samples wer-den 158 Haufen durch Beobachtungen im Optischen best¨atigt und deren photometrische Rotverschiebungen bestimmt. Wir finden eine Rotverschiebungsspanne von 0.1 . z . 1.35, was unsere Erwartung, daß Samples, die aus dem SZE–Signal selektiert wurden, ein großes Rotverschiebungsintervall abdecken, best¨atigt. Ein Vergleich zwischen der Signifikanz ξ der Haufenselektion auf Basis der SZE–Detektion und der Reinheit des Samples demonstriert, daß das SPT eine sehr saubere SZE–Auswahl liefert: Die Best¨atigungsrate des Samples betr¨agt 70% bei ξ > 4.5 und erreicht 100% bei ξ > 6. In 146 Haufen identifizieren wir die roten hellsten Haufengalaxien (red Brightest Cluster Galaxies, rBCGs) mittels der red-sequence–Technik. Wir vergleichen in unseren Haufen die r¨aumliche Verteilung der rBCGs im Bezug zum SZE– Haufenschwerpunkt mit der Verteilung eines anhand von R¨ontgenstrahlung ausgew¨ahlten Samples. Wenn die Methode zur Identifizierung der rBCGs ¨ahnlich ist, stellt sich heraus, daß auch die Verteilung der rBCGs beider Samples identisch ist. Da die rBCG–Verteilung durch Verschmelzung von Galaxien in der Haufenpopulation beeinflusst wird, bedeutet dies, daß Samples beider Auswahlmethoden (SZE und R¨ontgen) ¨ahnliche Indizien f¨ur kontinuierliche Akkretion liefern.

Anschließend analysieren wir die optischen Eigenschaften der ersten vier SZE–ausgew¨ahlten Haufen im Detail. Diese Haufen sind in einem f¨ur das gesamte SPT–Sample repr¨asentativen Rotverschiebungsintervall verteilt. F¨ur jeden Haufen verwenden wir aus R¨ontgenbeobachtun-gen gewonnene Massenabsch¨atzunR¨ontgenbeobachtun-gen und spektroskopische RotverschiebunR¨ontgenbeobachtun-gen, um die pro-jizierten Virialradien zu bestimmen. Wir finden Navarro-Frenk-White (NFW) Dichteprofile, die mit den Galaxienverteilungen kompatibel sind, und ermitteln Konzentrationsparameter ¨ahnlich derer r¨ontgenausgew¨ahlter Haufen. Des Weiteren studieren wir die Leuchtkraftfunk-tionen (LFs) der Haufen und vergleichen ihre charakteristischen Helligkeiten in jedem griz– Filter mit einem einfachen Sternenpopulationsmodell (Simple Stellar Population, SSP) und stellen deren ¨Ubereinstimmung fest. Um die Steigung α am schwachen Ende der Leuchtkraft-funktion besser bestimmen zu k¨onnen, nutzen wir die Voraussagen des SSP–Modells f¨ur die charakteristische Leuchtkraft als Funktion von Wellenl¨angenband und Rotverschiebung, und fitten eine Schechter–Funktion an die individuellen Leuchtkraftfunktionen der Haufen. Die Ergebnisse ergeben Werte f¨ur α, die konsistent sind mit den Werten, die in Studien publiziert wurden, welche auf Haufen basieren, die rein optisch oder anhand ihrer R¨ontgenstrahlung identifiziert wurden. Weiterhin bestimmen wir den Anteil blauer Galaxien innerhalb der

cher an blauen Galaxien sind als ihre Gegenst¨ucke bei niedrigen Rotverschiebungen. Die Fehler in diesen Messungen sind jedoch groß genug um auch kompatibel zu sein mit einem Szenario, in dem sich der Anteil der blauen Galaxien nicht mit der Rotverschiebung entwick-elt. Dieser Trend wird auch in anderen optischen Studien gefunden. Zusammenfassend zeigt unsere Pilotstudie, daß Galaxienverteilungen in SZE–ausgew¨ahlten Haufen sich nicht von den Verteilungen in durch andere Kriterien ausgew¨ahlten Haufen unterscheiden.

In Kapitel 5 pr¨asentieren wir eine systematische Studie der Entwicklung der Galaxienpop-ulation in den 26 massereichsten SZE-ausgew¨ahlten Haufen aus dem 2500 deg2

Beobach-tungsbereich des SPT. Wir erzeugen SSP–Modelle f¨ur jede Kombination von Teleskop und Filter aus den optischen Folgebeobachtungen. Wir kombinieren die radialen Profile einerseits unter Verwendung aller Galaxien innerhalb der Virialradien sowie unter ausschließlicher Ver-wendung der roten Galaxien und finden, daß die Verteilung der roten Galaxien geringf¨ugig konzentrierter ist als die Verteilung unter Ber¨ucksichtigung aller Galaxien. Weiterhin kom-binieren wir auch die Leuchtkraftfunktionen aller Galaxien und fitten eine Schechter–Funktion an das Ergebnis. Die resultierenden Werte f¨ur die charakteristischen Leuchtkr¨afte m∗_{, f¨ur die}

Galaxiendichte φ∗ _{bei ebendieser m}∗_{, sowie f¨ur α sind konsistent mit den Literaturwerten.}

Des Weiteren studieren wir die Entwicklung der Galaxienpopulation im Detail und finden, daß die Entwicklung von m∗ _{mit der Rotverschiebung in guter ¨Ubereinstimmung mit sich passiv}

entwickelnden SSP–Modellen ist. Wenn m∗ _{an das SSP–Modell gekoppelt wird, zeigt ein Fit}

an die Leuchtkraftfunktion keinen Trend in φ∗ _{und einen 2σ–Trend in α, wobei Haufen bei}

hoher Rotverschiebung ein flacheres α besitzen. Mit allen Parametern der Leuchtkraftfunk-tion zur Verf¨ugung k¨onnen wir die HalobesetungsfunkLeuchtkraftfunk-tion (Halo OccupaLeuchtkraftfunk-tion Number, HON) untersuchen. Wir fitten die HON mit einem Potenzgesetz N = N0 × (M/Mpivot)s(1 + z)γ,

wobei wir f¨ur die Steigung s die Literaturwerte ¨ubernehmen, da unser Haufensample nur einen kleinen Massenbereich abdeckt. Die gemessene Normalisierung N0 ist konsistent mit

der Normalisierung aus der Literatur bei gleicher Steigung. Weiterhin finden wir Hinweise auf Entwicklungstrends in der HON auf einem 2σ–Level, wobei SZE–ausgew¨ahlte Haufen bei hoher Rotverschiebung weniger Galaxien pro Masseneinheit haben als ihre Gegenst¨ucke bei niedrigen Rotverschiebungen.

Das Abschlußkapitel bildet eine Zusammenfassung der Ergebnisse dieser Doktorarbeit und skizziert zuk¨unftige Forschungsrichtungen.

Abstract

We present a study of the galaxy populations in massive galaxy clusters selected by their Sunyaev–Zel’dovich Effect (SZE) signatures. Selection via the SZE is approximately mass-limited where the mass limit varies only slightly with redshift, making it an ideal selection method for studying the evolution of the galaxy content of clusters. We begin by introducing the SZE, the South Pole Telescope (SPT), and the larger research project in which this Thesis is embedded. We then present the core galaxy population studies of this Thesis.

In Chapter 3, we present the first large-scale follow-up of an SZE-selected galaxy cluster sample. Of 224 galaxy cluster candidates in the sample, we optically confirm 158 clusters and measure their photometric redshifts. We find a redshift range of 0.1 . z . 1.35, confirming our expectation that samples selected by their SZE signal yield a large range of cluster redshifts. A comparison between the cluster SZE detection significance (ξ) and the purity of the sample demonstrates that the South Pole Telescope (SPT) produces a very clean SZE selection: the confirmation rate of the sample is 70% at ξ > 4.5 and reaches 100% at ξ > 6. In 146 clusters, we identify the red Brightest Cluster Galaxies (rBCG) using the red sequence technique. We compare the spatial distribution of the rBCGs with respect to the SZE cluster centroid in our clusters with that from an X-ray-selected sample. We find that if the method of identifying the rBCG is similar, the SZE rBCG distribution is indistinguishable from the rBCG distribution in an X-ray selected sample. Because the rBCG distribution is affected by merging in the cluster population, this indicates that SZE and X-ray selected samples exhibit similar evidence for continued accretion.

We then analyze the optical properties of the first four SZE-selected clusters in detail. These clusters are distributed in a redshift range representative of the entire SPT sample. For each cluster we use X-ray mass estimation and spectroscopic redshifts to define the projected virial radius. We then find Navarro-Frenk-White (NFW) profiles compatible with the galaxy distributions and recover concentration parameters similar to the values found for X-ray-selected clusters. We also study the cluster luminosity functions (LFs) and compare their characteristic luminosities in each griz band to a Simple Stellar Populations (SSP) model, finding them in agreement. To better constrain the faint end slope α, we adopt the SSP model prediction for the characteristic luminosity as a function of band and redshift and fit a Schechter function to the individual cluster LFs. The results reveal values of α that are consistent with those published for optically- and X-ray-selected clusters. We also measure the blue galaxy fraction within the virial region of each cluster, finding high-redshift clusters to be richer in blue galaxies than their low-redshift counterparts. However, the measurement errors from this study are large enough to be consistent with no evolution of blue galaxy fraction. This trend is similar to results found in other optical studies. In summary, our initial study shows that the galaxy populations of SZE-selected clusters do not differ from galaxy populations of clusters selected by other means.

In Chapter 5 we present a systematic evolutionary study of the galaxy populations in the 26 most massive SZE-selected clusters from the 2500 deg2_{SPT footprint. We create SSP models}

for each combination of telescope and bandpass used for the optical follow-up. We stack the radial profile using all of the galaxies within the virial radius as well as using only the red sequence galaxies. We find profiles from the latter galaxy set are slightly more concentrated.

The resulting values for the characteristic luminosity m∗_{, the density of galaxies φ}∗ _{at the}

characteristic m∗_{, and α are consistent with literature values. We then study the evolution of}

the galaxy population in detail and find that the redshift evolution of m∗ _{is in good agreement}

with the passively evolving SSP models. If m∗ _{is fixed to the SSP models, an LF fit reveals}

no trend for φ∗ _{and a 2σ trend for α, where clusters at high redshift have a shallower α.}

With all LF parameters in hand, we explore the Halo Occupation Number (HON). We fit the HON to the power law relation N = N0 × (M/Mpivot)s(1 + z)γ, fixing the slope s to

the literature value given the comparatively small range in mass in this cluster sample. The measured normalization N0 is consistent with the normalization found in the literature with

this same slope. We also find evidence for evolutionary trends in the HON at the 2σ level, where SZE selected high redshift clusters have fewer galaxies per unit mass than their low redshift counterparts.

In the concluding chapter, we provide a summary of the results presented in this Thesis and outline future directions of research.

Contents

Zusammenfassung ix

Abstract xi

Contents xv

List of Figures xviii

List of Tables xix

1 Introduction 1

2 Galaxy Clusters, the Sunyaev–Zel’dovich Effect, and the South Pole

Tele-scope 15

2.1 First Results: Method confirmation . . . 17

2.2 First Data Release Results . . . 19

2.3 Second Data Release Results . . . 21

2.4 Third Data Release Results . . . 22

2.5 X-ray Studies of SPT SZE selected Galaxy Clusters . . . 25

2.6 Weak Lensing Results . . . 29

2.7 Special clusters . . . 29

2.8 Submillimiter Galaxies . . . 31

2.9 Conclusion . . . 31

3 Redshifts, Sample Purity, and BCG Positions SPT Galaxy Clusters on 720 Square Degrees 33 3.1 Abstract . . . 33

3.2 Introduction . . . 34

3.3 Discovery & Followup . . . 35

3.3.1 SPT Data . . . 35

3.3.2 Optical/NIR Imaging . . . 36

3.3.3 Spectroscopic Observations . . . 39

3.4 Methodology . . . 39

3.4.3 Identifying rBCGs in SPT Clusters . . . 49

3.5 Results . . . 49

3.5.1 Redshift Distribution . . . 50

3.5.2 Purity of the SPT Cluster Candidates . . . 52

3.5.3 rBCG Offsets in SPT Clusters . . . 53

3.6 Conclusions . . . 57

3.7 Notable clusters . . . 59

4 A Multiband Study of the Galaxy Populations of the First Four SZ selected Galaxy Clusters 67 4.1 Abstract . . . 67

4.2 Introduction . . . 68

4.3 Observations and Data Reduction . . . 69

4.3.1 Blanco Cosmology Survey . . . 69

4.3.2 Completeness . . . 70

4.4 Basic Properties of these SPT Clusters . . . 72

4.4.1 Redshifts . . . 73

4.4.2 Cluster masses . . . 74

4.5 Cluster Galaxy Populations . . . 75

4.5.1 Radial distribution of galaxies . . . 75

4.5.2 Luminosity functions . . . 79

4.5.3 Halo Occupation Number . . . 85

4.5.4 Blue fractions . . . 87

4.6 Conclusions . . . 90

5 Galaxy Populations in the 26 most massive Galaxy Clusters in the South Pole Telescope SZE Survey 93 5.1 Abstract . . . 93

5.2 Introduction . . . 94

5.3 Observations and Data Reduction . . . 95

5.3.1 mm-wave Observations . . . 95

5.3.2 Redshifts and Cluster Masses . . . 96

5.3.3 Optical Imaging . . . 97

5.3.4 Completeness . . . 99

5.4 Cluster Galaxy Populations: tools . . . 99

5.4.1 Radial Distribution of Galaxies . . . 99

5.4.2 Luminosity Function . . . 101

5.4.3 Simple Stellar Population Models . . . 105

5.4.4 Simulated Catalogs . . . 105

5.5 Results . . . 107

5.5.1 Radial Profile . . . 107

5.5.2 Luminosity Function . . . 107

5.5.3 Halo Occupation Number . . . 112

5.6 Conclusions . . . 114

CONTENTS xv

Bibliography 142

Acknowledgments 143

List of Figures

1.1 Velocity-Distance relation as discovered by Hubble. . . 6

1.2 Redshift-distance relationship fit for different cosmological models. . . 6

1.3 ESA/Planck Universe Energy content . . . 7

1.4 Planck CMB temperature map. . . 8

1.5 Planck CMB Angular power spectrum. . . 8

1.6 BAO from Eisenstein et al. (2005) . . . 10

1.7 Vikhlinin et al. (2009) mass function . . . 11

1.8 Bullet Cluster . . . 12

1.9 Combination of different cosmological probes . . . 13

2.1 Distortion in the CMB due to the thermal SZE. . . 16

2.2 The South Pole Telescope. . . 17

2.3 CMB decrement of the first blind SZE cluster detections. . . 18

2.4 Optical images of the first four SZE selected clusters with the SPT . . . 19

2.5 First cosmology analysis on a SZE selected cluster sample. . . 20

2.6 Richness in SZE-selected Clusters . . . 21

2.7 Rarity of the 26 most massive galaxy clusters in the SPT footprint. . . 23

2.8 YSZ− YX relation from Andersson et al. (2011). . . 26

2.9 YSZ− M500 relation from Andersson et al. (2011). . . 26

2.10 Projected BCG / X–ray centroid offset as function of the surface brightness concentration. . . 27

2.11 BCG in a cool core cluster with massive star formation. . . 28

2.12 Redshift evolution of cooling properties. . . 30

3.1 Photometric versus spectroscopic redshift for 47 confirmed SZ-selected clusters. 43 3.2 Weighted mean photometric redshift versus spectroscopic redshift. . . 46

3.3 Photometric versus spectroscopic redshift from Spitzer IRAC 3.6µm-4.5µm colors. 47 3.4 Redshift histogram of 158 confirmed clusters. . . 50

3.5 Purity estimates from the optical/NIR followup compared to simulations. . . 51

3.6 rBCG / SPT candidate position offsets for 146 systems. . . 54

3.7 Normalized cumulative distribution of rBCG / SPT candidate position offsets. 56 4.1 Completeness of the BCS coadds for the SPT-CL J0516-5430 field. . . 71

4.4 m∗ _{resulting from Schechter function fits to the luminosity function. . . . 78}

4.5 Luminosity function with best fit Schechter function for SPT-CL J0516-5430. 81 4.6 Luminosity function with best fit Schechter function for SPT-CL J0509-5342. 81 4.7 Luminosity function with best fit Schechter function for SPT-CL J0528-5300. 82 4.8 Luminosity function with best fit Schechter function for SPT-CL J0546-5345. 82 4.9 68% confidence region for the LF parameters for each cluster / band combination. 83 4.10 Halo occupation number within each band for each cluster. . . 86

4.11 Color magnitude diagram for galaxies around each cluster. . . 88

4.12 Blue fraction versus redshift using the populations from Fig. 4.11. . . 90

5.1 Mass-redshift distribution of Williamson et al. (2011) sample. . . 95

5.2 Stacked radial profiles. . . 102

5.3 Evolution of the concentration parameter. . . 103

5.4 Individual and stacked luminosity function. . . 108

5.5 Evolution of the luminosity function parameters. . . 111

List of Tables

2.1 Mass Scaling Relations . . . 22

2.2 R12 Cosmological constraints . . . 24

3.1 Optical and infrared imagers . . . 37

3.2 Spectroscopic Follow-Up . . . 40

3.3 All candidates above ξ = 4.5 in 720 deg2 _{of the SPT-SZ survey. . . . 61}

4.1 Completeness limits for each tile for each filter for 90%/50% completeness . . 71

4.2 X-ray masses, spectroscopic redshifts and cluster parameters. . . 74

4.3 HON parameters . . . 84

5.1 SPT Cluster List. . . 96

Chapter

1

Introduction

The beginning of the 20th_{century was an exciting time. Along with a new theory of gravitation}

(Einstein, 1916) and its non-static solutions (Friedmann, 1922; Lemaˆıtre, 1931; Robertson, 1933), the works of astronomers such as Vesto Slipher and Edwin Hubble revealed a Universe greater and more dynamic than their predecessors imagined. The observational proof that fuzzy objects known as nebulae were in fact new kinds of objects that lay beyond the Milky Way Galaxy produced a new picture of our Universe (Slipher, 1913; Stromberg, 1925; Hubble, 1929). With this discovery a fundamental change took place: the long-standing paradigm of a static Universe changed to one that is a dynamic and brought forth an entire new discipline in extragalactic astronomy.

The first catalogs of galaxies date back to the 18th _{century with the work of Charles Messier}

(Messier, 1781) and F. Wilhelm Herschel (Herschel, 1785). From Paris, Messier recorded the positions and diameters of “star clusters” and what were back then referred to as nebulae. From England, Wilhelm Herschel discovered thousands of nebulae and recognized several nearby galaxy clusters and groups. Herschel and Messier were the first to recognize a concen-tration of nebulae in the direction of the Virgo and Coma clusters, which would later become become the most studied galaxy clusters in astronomy. There is no greater example of the impact that these clusters had on astronomy than the work of Fritz Zwicky on the Coma cluster in 1933. Zwicky used spectroscopic observations to find the motions of the galaxies in order to estimate the mass of the cluster. The results revealed for the first time the need for some form of dark matter to explain the high velocities of galaxies at large radii (Zwicky, 1933). Since then, large samples of galaxy clusters have been compiled (e.g. Abell, 1958; Zwicky et al., 1968). Even early on, clusters of galaxies acquired an important status for our understanding of the Universe in general and cosmology in particular.

Galaxy Clusters Components

Galaxy clusters are the largest gravitationally collapsed structures, consisting of hundreds to thousands of galaxies and masses of 1014_{M− 10}15_{M. Although clusters were recognized}

early on due to their remarkable agglomeration of objects bright in optical light, stars consti-tute only about 2-4% of the Cluster mass. Intracluster gas only contributes between 8-15% to the total cluster mass (Gonzalez et al., 2013). The majority of the cluster mass is neither in form of stars, gas, nor any other form of baryonic matter. The remaining 85% of the mass

cluster components in detail below.

Galaxies

Clusters are unique places to study galaxies because they provide a volume-limited sample of galaxies. The two first clusters to have their galaxy populations characterized were the Coma and Virgo clusters. The galaxy luminosity segregation in both clusters was reported in several papers (Zwicky, 1942, 1951; Reaves, 1966; Rood & Turnrose, 1968; Rood, 1969; Rood & Abell, 1973) Here, the brightest galaxies were found in the cores of the clusters and the general population luminosity was found to diminished with cluster-centric radius (Oemler, 1974). The works of Rood (1969) and Rood & Abell (1973) showed the importance of the luminosity function and the color-magnitude diagram as statistical tools to look at the overall changes in the galaxy populations as well as to understand the formation of the stellar component. The luminosity function describes the number of galaxies per luminosity bin and therefore informs us about the galaxy populations in a statistical manner. Examples of luminosity functions are shown Fig. 4.5 and Fig. 5.4.

When plotting a cluster’s member galaxies on a color-magnitude diagram (CMD) a sequence of red galaxies forms a clearly-defined ridge. The placement of this ridge is determined by the redshifted location of the 4000˚A break in typical galaxy spectra (Baum, 1959; Rood, 1969; Visvanathan & Sandage, 1977) as well as the collective age of the stars in the member galaxies. The tight scatter of the red sequence in the CMD suggests a single, short formation epoch of the cluster’s stellar component at very high redshift (z & 2 − 3) that is collectively aging (e.g. De Lucia et al., 2004; Rudnick et al., 2012). The red sequence slope or tilt is driven by the metallicity, with larger and brighter galaxies redder (higher metallicity) as their deeper potential wells allows them to retain more metals. Several examples of the red sequence at various redshift using different colors is shown in Fig. 4.11.

Galaxies also provide a means to estimate the cluster mass. The most direct measurement is via the cluster’s line-of-sight velocity dispersion. Assuming the galaxies are dynamically relaxed, they act as collisionless particles tracing the cluster’s gravitational potential. The method provides a direct estimation of the total cluster mass with measurement errors on the order of ≈ 20 %, as long as systematics are kept under control (Saro et al., 2013).

The number of galaxies in a cluster, (known as the cluster’s richness), can also used to estimate the cluster’s mass. However, this method first requires calibration on a set of clusters with known mass and is therefore dependent on other mass estimation techniques. Furthermore the resulting measurement errors are comparatively large. A richness versus mass plot is shown in Fig. 2.6.

Intracluster light

The first detection of Intra-cluster light (ICL) dates to 1952, when Zwicky found intergalactic stars and groups of stars in the Coma cluster (Zwicky, 1952). A general definition of ICL is that it is light from stars not bound to identified galaxies. Simulations have shown that this diffuse stellar component comes from tidal stripping during galaxy mergers. Typically half of these stars come from the Brightest Cluster Galaxy (BCG), a quarter come from tidal interaction of less massive galaxies, and the remaining quarter come from dissolved galaxies (e.g. Murante et al., 2007). The majority of the observable ICL is found near the core of the

3 cluster, where the BCG commonly resides. Therefore, in galaxy clusters where the BCG sits near the center, the BCG and ICL must be decomposed. A two-component model for the BCG and the ICL is found to be a good fit (e.g. Gonzalez et al., 2005). In such cases, the ICL represents between 80-90% of the total light of the two components within 300 kpc.

Gas component

After the suggestion that hot, intracluster gas must fill clusters (Limber, 1959), Felten et al. (1966) estimated that a thermalized gas in the Coma cluster would have a temperature around ≈ 7 × 107_{K. The source of this heat comes from adiabatic collapse of the gas in}

the cluster’s deep potential well. At these temperatures, the gas is fully ionized and free electrons suffer Coulomb interactions with the ions, emitting x-rays in the keV energy range (known as thermal bremsstrahlung radiation). This x-ray emission was first observed by the Uhurusatellite from the Aerobee 150 rocket Data in the 2-8 keV range from Uhuru revealed extended emission in rich clusters with luminosities of 1043_{− 10}44 _{ergs s}−1 _{(Gursky et al.,}

1972). The evidence of the x-ray emission as thermal bremsstrahlung became stronger as the x-ray luminosity was found to be highly correlated with cluster velocity dispersion, pointing to a connection to the size of the cluster potential through the virial theorem (Solinger & Tucker, 1972).

Under the assumption of hydrostatic equilibrium, the gas temperature provides a good esti-mate of the cluster virial mass via the relation M ∝ Tgas3/2. Furthermore, the x-ray luminosity

can be used as a mass proxy as it is connected to the gas temperature as LX∝ Tgas2 (Giodini

et al., 2013). A third way to measure the cluster mass using the intracluster gas is to measure how the hot electrons distort Cosmic Microwave Background photons by inverse Compton scattering. This phenomenon is known as the Sunyaev–Zel’dovich Effect (SZE) and discussed in more detail in Chapter 2.

Dark Matter

The first evidence of Dark Matter (DM) in the Universe appeared from spectroscopic obser-vations of the Coma cluster. Zwicky (1933) used the virial theorem to argue that the velocity dispersion of galaxies in the Coma cluster, observed to be on the order of 1500 to 2000 km/sec, indicates that the cluster must have an average density of 400 times the mass derived from observations of the luminous matter. Additional evidence of DM came from observations of galaxy rotations curves. Babcock (1939) studied the kinematics of the Andromeda galaxy noticing that rotation curves approached a constant angular velocity in the outer spiral arms. Babcock (1939) also reported that such velocities would imply a mass-to-light ratio of about 50, attributing the large mass estimation to “the outer parts of the spiral on the basis of the unexpectedly large circular velocities of these parts”. Further evidence based on galaxy rotation curves strengthens the need for DM in galaxies (e.g. Seielstad & Whiteoak, 1965; Roberts, 1969; Rubin & Ford, 1970).

The possibility that the unseen DM in galaxies takes a form of baryonic dark objects was explored by several projects such as MACHO, OGLE, and EROS. They searched for MAssive Compact Halo Objects (MACHO) like brown dwarfs and planets to test if their abundance could explain the observed missing mass. MACHOs can be detected from the magnification of bright objects as they cross our line-of-sight, due to gravitational lensing. Several searches were carried out, using the Andromeda Galaxy, the Galactic bulge, and the Magellanic clouds

results only account for ≈ 20% of the missing mass (Alcock et al., 1997; Udalski et al., 1997; Alcock et al., 2000; Afonso et al., 2003; Tisserand et al., 2007; Garg, 2008).

With the list of baryonic candidates exhausted, other forms of matter are explored. Some of the DM candidates include Weakly Interacting Massive Particles (WIMPs), Axions, nonther-mal WIMPS or WIMPzillas, Q-balls, gravitinos, etc. Whatever their true nature, nonbaryonic DM particles would at least have to be collisionless, cold, and behave like a fluid.

In recent investigations, galaxy clusters have provided striking evidence of the nature of DM. This is achieved by measuring the gravitational lensing of background galaxies due to the space-time distortions from a massive, foreground galaxy cluster. A statistical analysis of the distorted shapes of the background galaxies allows us to extract the total mass of the foreground cluster. Because DM is the dominant source of cluster mass, the distortions are assumed to be caused primarily by DM. Therefore this kind of analysis can reveal how the DM is distributed within a cluster. One of the most famous exhibits of this technique is the Bullet Cluster, a system where two clusters are colliding. By complementing the background-galaxy shape measurements with x-ray observations, the three main components of the cluster are traced: x-rays follow the gas, shape measurements of background galaxies indicate where the bulk of the (dark) matter is, and optical images shows where the galaxies are located. The three different components are depicted in Fig. 1.8. Here, it is clearly shown that the DM distribution derived from shape measurements of background galaxies does not follow the collisional gas component, but rather the collisionless component traced by the galaxies.

Galaxy Clusters and other Cosmological Probes

Once the dynamical nature of the Universe was established, its understanding and character-ization became a goal. This required new, cosmological probes, such as galaxies (e.g. Hubble, 1929; Lemaˆıtre, 1931; Bender et al., 1998), galaxy clusters (e.g. Wang & Steinhardt, 1998; Haiman et al., 2001; Reichardt et al., 2013), supernova type Ia (e.g. Hamuy et al., 1993; Schmidt et al., 1998; Riess et al., 1998; Perlmutter et al., 1999), the Cosmic Microwave Back-ground (CMB) (e.g. Smoot et al., 1992; Komatsu et al., 2011; Planck Collaboration et al., 2013), and Baryon Acoustic Oscillations (e.g. Eisenstein & Hu, 1998; S´anchez et al., 2012). Utilizing multiple probes is important as each cosmological probe reveals different aspects of the evolution of the Universe and each is subject to different systematics. Together, they form the current picture of cosmology and is the context within which this thesis was developed. In this chapter we introduce the cosmological probes used in this thesis.

Type Ia Supernova

In 1998, two teams lead by Adam Riess (Riess et al., 1998) and Saul Perlmutter (Perlmutter et al., 1999) searched for the rate of deceleration of the universe, using a set of high redshift supernovae (SNe). However, instead of a decelerating Universe, they discovered that the Uni-verse is actually accelerating.

To trace the dynamics of the universe, distances and redshifts of objects spanning a range of distances are needed. While the redshift can be obtained precisely from the object’s spectra, the distance is more tricky. To determine the distance, scientists use a class of objects with known intrinsic luminosity to serve as standard candles to which distances can be measured.

5 Some examples of standard candles are Cepheids (used by Hubble, see Feast & Walker, 1987, for a review), Planetary Nebulae (e.g. Ciardullo et al., 2002, and references therein) and Type Ia Supernovae (SNIa). Because brighter objects can be observed further away, the greater the luminosity of a standard candle, the greater the importance for tracing the dynamics of the Universe. Among all known standard candles, SNIa are the most luminous, with a peak observed energy of ∼ 1043_{ergs/sec. This is so bright that it outshines its host galaxy for a}

few days, providing us a brief window for observation from very far away.

Although SNe are extremely luminous events, they are not exactly standard candles, as their peak luminosities have intrinsic scatter of about ≈ ±0.7 in BVI. The key is to make them a standard candle by correcting their peak luminosity. Phillips (1993) found that the decay in magnitude that a SNIa suffers within 15 days is highly correlated with the absolute magnitude at the peak of the light curve. Using this information the peak luminosity can be corrected and therefore the absolute magnitude can be standardized. More sophisticated methods have been developed (e.g. Multicolor Light Curve Shape Method (MLCS), “stretch factor”; Riess et al., 1998; Perlmutter et al., 1999) but they share the same principle.

Once the SNIa are calibrated and their redshifts are determined, it is possible to find out how the Universe is evolving applying the following formula to a set of observed standard candles:

m − M = −2.5log  L 4πd2 L  + 2.5log  L 4π(10pc)2  = 5log_10pcdL , (1.1) where dL is the luminosity distance which is a function of redshift and chosen cosmology. In

the local Universe dL can be described simply as cz/H0, where H0 is the Hubble parameter

and corresponds to the rate of expansion of the Universe. At cosmological scales dLfor a flat

Universe is expressed as dL= (1 + z)2DA(z) = c(1 + z) Z z 0 dz0 H(z0₎ = c(1 + z) H0 Z z 0 dz0 ΩM(1 + z0)3+ ΩΛ. (1.2)

In such a Universe, ΩM is the matter energy density at redshift zero and ΩΛ is an unknown

energy component (Dark Energy). Thus, eq. 1.1 becomes: m − M = 5logc(1 + z)/H0 Rz 0 dz 0 ΩM(1+z0)3+ΩΛ 10pc (1.3)

Fig. 1.2 shows the data presented in Riess et al. (1998) along with eq. 1.3 evaluated for two closed Universes, a pure matter-dominated one and a pure Λ-dominated one, as well as a fiducial, open Universe with only matter (ΩM = 0.2 and ΩΛ = 0.0). All curves use a

Hubble parameter of H0 = 65.2. The results clearly show that we are not living in a open,

matter-dominated Universe! For a review see Clocchiatti (2011).

The Cosmic Microwave Background

The Cosmic Microwave Background (CMB) radiation is the light that has freely streamed since a redshift of ∼1100, when the Universe first cooled to a temperature below 3000 K. At that temperature, photons were no longer energetic enough to keep hydrogen and helium atoms ionized, and the two atoms recombine. This period is also known as the epoch of

6 CHAPTER 1. INTRODUCTION Secondly, the scatter of the individual nebulae can be examined by assuming the relation between distances and velocities as previously determined. Distances can then be calculated from the velocities corrected for solar motion, and absolute magnitudes can be derived from the apparent magnitudes. The results are given in table 2 and may be compared with the distribution of absolute magnitudes among the nebulae in table 1, whose distances are derived from other criteria. N. G. C. 404 can be excluded, since the observed velocity is so small that the peculiar motion must be large in comparison with the distance effect. The object is not necessarily an exception, however, since a distance can be assigned for which the peculiar motion and the absolute magnitude are both within the range previously determined. The two mean magnitudes, -15.3 and -15.5, the ranges, 4.9 and 5.0 mag., and the frequency distributions are closely similar for these two entirely independent sets of data; and even the slight difference in mean magnitudes can be attributed to the selected, very bright, nebulae in the Virgo Cluster. This entirely unforced agreement supports the validity of the velocity-distance relation in a very evident matter. Finally, it is worth recording that the frequency distribution of absolute magnitudes in the two tables combined is comparable with those found in the various clusters of nebulae.

Velocity-Distance Relation among Extra-Galactic Nebulae.

Figure 1: Radial velocities, corrected for solar motion, are plotted against distances estimated

from involved stars and mean luminosities of nebulae in a cluster. The black discs and full line represent the solution for solar motion using the nebulae individually; the circles and broken line represent the solution combining the nebulae into groups; the cross represents the mean velocity corresponding to the mean distance of 22 nebulae whose distances could not be estimated individually.

The results establish a roughly linear relation between velocities and distances among nebulae for which velocities have been previously published, and the relation appears to dominate the

distribution of velocities. In order to investigate the matter on a much larger scale, Mr. Humason at Mount Wilson has initiated a program of determining velocities of the most distant nebulae that can Figure 1.1: Diagram showing the nature of the dynamical universe by Hubble. The scientific concept of the Universe as an eternal and immutable entity gives away to a dynamical (and more interesting) Universe.

-0.5 0 0.5 0.01 0.1 1 ∆ (m-M) redshift Ω_M=0.2, Ω_Λ =0.0 Ω_M=0.24, Ω_Λ =0.76 Ω_M=1.0, Ω_Λ =0.0 Riess et al. (1998) MLCS calibrated SNe

Figure 1.2: ∆m from Riess et al. (1998). The best fit for a flat cosmology is the solid line with H0 = 65.2,

7

Figure 1.3: Composition of the universe according to the ESA/Planck mission results. Credit: ESA/Planck

the last scatter. Before the last scattering the photons were in equilibrium and produced a blackbody spectrum with a temperature measured today of 2.73 K (Fixsen et al., 1996). This last scattering radiation was predicted as early as 1948 (Alpher et al., 1948) and ob-served in McKellar (1941, without realizing) and Penzias & Wilson (1965, without intending). Although highly homogeneous, the CMB has anisotropies at the ∆T/T ∼ 10−5 _{level (Smoot}

et al., 1992). Between the time of inflation and last scattering, the Baryonic matter was ionized and coupled to the radiation. In this state, it behaved as a fluid and supported sound waves (Peebles & Yu, 1970; Sunyaev & Zel’dovich, 1970). However, DM was not coupled and began to clump together, forming deeper and deeper gravitational potentials and planting the primordial seeds for the large-scale structure of the present day. The photon-baryon fluid interacts with the DM potential wells, where the battle between gravity and pressure causes the baryons to oscillate, creating acoustic waves. The wavelengths of these waves are har-monics of the fundamental scale set by the distance that the sound waves travel in the time between the formation of the potential wells and recombination.

Statistically those waves at maximum compression and rarefaction are the acoustic peaks that we observe in the CMB angular power spectrum of the temperature maps as traced by the light (Planck Collaboration et al., 2013, see Fig. 1.5). The heights and positions of such acoustic peaks provide insights into the content and geometry of the Universe. Predictions from varying cosmologies can be generated using numerical codes such as CMBFAST (Seljak & Zaldarriaga, 1996) or CAMB (Lewis et al., 2000). The results can be compared to observations to discern the validity of the tested cosmology. In this way, CMB experiments like WMAP (Komatsu et al., 2011; Bennett et al., 2012) and Planck (Planck Collaboration et al., 2013) have rendered a detailed description of the Universe we live in.

Baryon Acoustic Oscillations

The Baryon Acoustic Oscillations describe the characteristic length of the first acoustic peak as it reaches recombination at the speed of sound. After recombination, the photons travel

Figure 1.4: CMB temperature map. Credit: ESA/Planck. Planck Collaboration: Cosmological parameters

Fig. 1. Planck foreground-subtracted temperature power spectrum (with foreground and other “nuisance” parameters fixed to their best-fit values for the base ΛCDM model). The power spectrum at low multipoles (� = 2–49, plotted on a logarithmic multi-pole scale) is determined by the Commander algorithm applied to the Planck maps in the frequency range 30–353 GHz over 91% of the sky. This is used to construct a low-multipole temperature likelihood using a Blackwell-Rao estimator, as described inPlanck Collaboration XV(2013). The asymmetric error bars show 68% confidence limits and include the contribution from un-certainties in foreground subtraction. At multipoles 50 ≤ � ≤ 2500 (plotted on a linear multipole scale) we show the best-fit CMB spectrum computed from the CamSpec likelihood (seePlanck Collaboration XV 2013) after removal of unresolved foreground com-ponents. The light grey points show the power spectrum multipole-by-multipole. The blue points show averages in bands of width ∆� ≈ 31 together with 1σ errors computed from the diagonal components of the band-averaged covariance matrix (which includes contributions from beam and foreground uncertainties). The red line shows the temperature spectrum for the best-fit base ΛCDM cosmology. The lower panel shows the power spectrum residuals with respect to this theoretical model. The green lines show the ±1σ errors on the individual power spectrum estimates at high multipoles computed from the CamSpec covariance matrix. Note the change in vertical scale in the lower panel at � = 50.

3 Figure 1.5: Planck Temperature power spectrum (Planck Collaboration et al., 2013). Credit: ESA/Planck

9 freely, ending the wave propagation and freezing the baryon overdensity at a characteristic length. The baryon overdensity grows over cosmological time, creating gravitationally col-lapsed systems such as galaxies. A complete, all-sky map of galaxies can be traced using the correlation function enhancement at a distance given by (e.g. Eisenstein & Hu, 1998; Percival et al., 2007): rs(z∗) = 1 H0Ω1/2_M Z a∗ 0 cs (a + aeq)1/2da,

where csis the sound speed, a is the scale factor (1+z)−1, a∗is the value at recombination and

aeq is the value at matter-radiation equality. This distance (∆χ) is around 150 (comoving)

Mpc in current cosmology and can be connected to observables through ∆θ = _d∆χ

A(z),

where ∆θ is the angular size in the sky. This in turn can be connected to the luminotity distance via the angular distance:

dA(z) = _{(1 + z)}dL ₂,

where dL is given in eq. 1.2. As the evolution of the standard ruler is a function of the

expansion rate, H(z), and the two redshifts at which the distances were measured, we can write:

c∆z = H(z)∆χ.

In this way, from purely geometrical considerations, we can measure H(z) which depends on cosmological parameters as follows:

H2_(z)

H2

0 = ΩM(1 + z) 3_{+ Ω}

r(1 + z)4+ Ωk(1 + z)2+ ΩΛ(1 + z)−3(1+w)

This geometrical test has been sucessfully applied: By measuring the large-scale correlation function on a spectroscopic sample of 46,748 Luminous Red Galaxies (LRG) from the Sloan Digital Sky Survey with a typical redshift of 0.35, Eisenstein et al. (2005) presented the first clear detection of the acoustic peak (see Fig. 1.6) along with its cosmological implications. An updated version can be found in S´anchez et al. (2012).

Galaxy Clusters

Galaxy clusters are the largest collapsed objects in the Universe. Hundreds to thousands of galaxies crowd together in deep potential wells, largely defined by Dark Matter (see Fig. 1.8). In this way, galaxy clusters trace the distribution of the mass in the Universe as well as the position of the initial density fluctuations. Because of these characteristics, galaxy clusters are a powerful probe for cosmological studies (Albrecht et al., 2006).

In the cosmological context, galaxy clusters are sensitive to the rate of the formation of structures which depends on the dynamical behaviour and mass content of the Universe. By simply using the abundance and the redshift distribution of clusters of galaxies as a function of mass, several cosmological parameters can be extracted (e.g. Haiman et al., 2001; Holder et al., 2001). However, to extract cosmological parameters, the observed redshift evolution of

perform these surveys by a factor of 2 in fractional errors on large scales. Note that quasar surveys cover much more volume than even the LRG survey, but their effective volumes are worse, even on large scales, due to shot noise.

3. THE REDSHIFT-SPACE CORRELATION FUNCTION 3.1. Correlation Function Estimation

In this paper, we analyze the large-scale clustering using the two-point correlation function ( Peebles 1980,_{x 71). In recent} years, the power spectrum has become the common choice on large scales, as the power in different Fourier modes of the linear density field is statistically independent in standard cosmology theories ( Bardeen et al. 1986). However, this advantage breaks down on small scales due to nonlinear structure formation, while on large scales elaborate methods are required to recover the sta-tistical independence in the face of survey boundary effects (for discussion, see Tegmark et al. 1998). The power spectrum and correlation function contain the same information in principle, as they are Fourier transforms of one another. The property of the independence of different Fourier modes is not lost in real space, but rather it is encoded into the off-diagonal elements of the covariance matrix via a linear basis transformation. One must therefore accurately track the full covariance matrix to use the correlation function properly, but this is feasible. An advantage of the correlation function is that, unlike in the power spectrum, small-scale effects such as shot noise and intrahalo astrophysics stay on small scales, well separated from the linear regime fluc-tuations and acoustic effects.

We compute the redshift-space correlation function using the Landy-Szalay estimator ( Landy & Szalay 1993). Random catalogs containing at least 16 times as many galaxies as the LRG sample were constructed according to the radial and angular se-lection functions described above. We assume a flat cosmology with!m¼ 0:3 and !"¼ 0:7 when computing the correlation

function. We place each data point in its comoving coordinate location based on its redshift and compute the comoving sep-aration between two points using the vector difference. We use bins in separations of 4 h#1Mpc from 10 to 30 h#1Mpc and bins of 10 h#1Mpc thereafter out to 180 h#1Mpc, for a total of 20 bins.

We weight the sample using a scale-independent weighting that depends on redshift. When computing the correlation func-tion, each galaxy and random point is weighted by 1/_{½1 þ n(z)P}w&

( Feldman et al. 1994), where n(z) is the comoving number density and Pw¼ 40;000 h#3 Mpc3. We do not allow Pwto change with

scale so as to avoid scale-dependent changes in the effective bias caused by differential changes in the sample redshift. Our choice of Pwis close to optimal at k' 0:05 h Mpc#1and within 5% of

the optimal errors for all scales relevant to the acoustic oscillations (kP 0:15 h Mpc#1). At z< 0:36, nPwis about 4, while nPw' 1

at z_{¼ 0:47. Our results do not depend on the value of P}w;

chang-ing the value wildly alters our best-fit results by only 0.1!. Redshift distortions cause the redshift-space correlation func-tion to vary according to the angle between the separafunc-tion vector and the line of sight. To ease comparison to theory, we focus on the spherically averaged correlation function. Because of the boundary of the survey, the number of possible tangential sep-arations is somewhat underrepresented compared to the number of possible line-of-sight separations, particularly at very large scales. To correct for this, we compute the correlation functions in four angular bins. The effects of redshift distortions are ob-vious: large-separation correlations are smaller along the

line-of-sight direction than along the tangential direction. We sum these four correlation functions in the proportions corresponding to the fraction of the sphere included in the angular bin, thereby re-covering the spherically averaged redshift-space correlation func-tion. We have not yet explored the cosmological implications of the anisotropy of the correlation function ( Matsubara & Szalay 2003).

The resulting redshift-space correlation function is shown in Figure 2. A more convenient view is shown in Figure 3, where we have multiplied by the square of the separation, so as to flatten out the result. The errors and overlaid models will be discussed below. The bump at 100 h#1Mpc is the acoustic peak, to be de-scribed in_{x 4.1.}

The clustering bias of LRGs is known to be a strong function of luminosity ( Hogg et al. 2003; Eisenstein et al. 2005; Zehavi et al. 2005a), and while the LRG sample is nearly volume-limited out to z_{! 0:36, the flux cut does produce a varying luminosity} cut at higher redshifts. If larger scale correlations were prefer-entially drawn from higher redshift, we would have a differential bias (see discussion in Tegmark et al. 2004a). However, Zehavi et al. (2005a) have studied the clustering amplitude in the two limiting cases, namely the luminosity threshold at z< 0:36 and that at z_{¼ 0:47. The differential bias between these two samples} on large scales is modest, only 15%. We make a simple param-eterization of the bias as a function of redshift and then compute b2 averaged as a function of scale over the pair counts in the random catalog. The bias varies by less than 0.5% as a function of scale, and so we conclude that there is no effect of a possible correlation of scale with redshift. This test also shows that the

Fig. 2.—Large-scale redshift-space correlation function of the SDSS LRG sample. The error bars are from the diagonal elements of the mock-catalog co-variance matrix; however, the points are correlated. Note that the vertical axis mixes logarithmic and linear scalings. The inset shows an expanded view with a linear vertical axis. The models are!mh2¼ 0:12 (top line), 0.13 (second line),

and 0.14 (third line), all with!bh2¼ 0:024 and n ¼ 0:98 and with a mild

non-linear prescription folded in. The bottom line shows a pure CDM model (!mh2¼

0:105), which lacks the acoustic peak. It is interesting to note that although the data appear higher than the models, the covariance between the points is soft as regards overall shifts in"(s). Subtracting 0.002 from "(s) at all scales makes the plot look cosmetically perfect but changes the best-fit#2_{by only 1.3. The bump}

at 100 h#1_{Mpc scale, on the other hand, is statistically significant. [See the electronic}

edition of the Journal for a colorversion of this figure.]

Figure 1.6: From Eisenstein et al. (2005):Large-scale redshift-space two-point correlation function of the SDSS Luminous Red Galaxy sample. The Baryon Acoustic Peak detection shown here confirms the theoretical prediction.

the cluster mass function per volume surveyed must be compared to its theoretical prediction for a given cosmology (e.g. Vikhlinin et al., 2009, see Fig. 1.7). An analytical description of the predicted number density of collapsed objects of a mass M at a redshift z in the linear regime is provided by the Press-Schechter mass function (Press & Schechter, 1974):

dn dM = − r 2 π ¯ ρm M δc σ2 M dσM dM exp(− δ2 c 2σ2 M),

where ¯ρm is the mean matter density, δc is the linear density contrast (1.686 for spherical

collapse) and σM is the variance of the density fluctuation field at scale M (see eq. 1.6).

Currently, the mass function that is used for cluster cosmology is derived from numerical simulations, such as the Tinker et al. (2008) mass function:

dn dM = f(σM)ρm M dlnσ−1_M dM , (1.4) with f(σM) parametrized as f(σM) = A  σ_M b −a + 1  e−c/σ2 M, (1.5)

where the values for A, a, b, and c are 0.186×(1+z)−0.14_{, 1.47×(1+z)}−0.06_{, 2.57×(1+z)}−0.011

and 1.19, respectively (−0.011 was obtained by evaluating 10−(0.75/log10(∆/75.))1.2

assuming M = M∆,mean= M200,mean). Also, σM is σ2_M= Z P (k) ˆW (kR)k2dk, (1.6)

11

4

VIKHLININ ET AL.

z = 0.55 − 0.90

z = 0.025 − 0.25

10

14

10

15

10

−

9

10

−

8

10

−

7

10

−

6

10

−

5

M

500

, h

−

1

M

"

N

(>M

),

h

−3

Mp

c

−3

Ω

_M

_{= 0.25, Ω}

_Λ

_{= 0.75, h = 0.72}

z = 0.025 − 0.25

10

14

10

15

10

−

9

10

−

8

10

−

7

10

−

6

10

−

5

M

₅₀₀

, h

−

1

M

"

N

(>M

),

h

−3

Mp

c

−3

Ω

_M

_{= 0.25,}

Ω

_Λ

= 0,

_{h = 0.72}

z = 0.55 − 0.90

F��. 2.— Illustration of sensitivity of the cluster mass function to the cosmological model. In the le� panel, we show the measured mass function and predicted

models (with only the overall normalization at z = 0 adjusted) computed for a cosmology which is close to our best-�t model. �e low-z mass function is reproduced

from Fig. 1, which for the high-z cluster we show only the most distant subsample (z > 0.55) to better illustrate the e�ects. In the right panel, both the data and the

models are computed for a cosmology with ΩΛ = 0. Both the model and the data at high redshi�s are changed relative to the ΩΛ = 0.75 case. �e measured mass

function is changed because it is derived for a di�erent distance-redshi� relation. �e model is changed because the predicted growth of structure and overdensity

thresholds corresponding to ∆crit = 500 are di�erent. When the overall model normalization is adjusted to the low-z mass function, the predicted number density

of z > 0.55 clusters is in strong disagreement with the data, and therefore this combination of ΩM and ΩΛ can be rejected.

of interest in our study; at this level, the theoretical

uncertain-ties in the mass function do not contribute signi�cantly to the

systematic error budget. Although the formula has been

cali-brated using dissipationless N-body simulations (i.e. without

e�ects of baryons), the expected e�ect of the internal

redistri-bution of mass during baryon dissipation on halo mass

func-tion are expected to be

< 5% (Rudd et al. 2008) for a realistic

fraction of baryons that condenses to form galaxies.

Similarly to Jenkins et al. (2001) and Warren et al. (2006), the

Tinker et al. formulas for the halo mass function are presented

as a function of variance of the density �eld on a mass scale M.

�e variance, in turn, depends on the linear power spectrum of

the cosmological model, P

(k), which we calculate as a product

of the initial power law spectrum, k

_{, and the transfer}

func-tion for the given mixture of CDM and baryons, computed

using the analytic approximations of Eisenstein & Hu (1999).

�is analytic approximation is accurate to better than 2% for

a wide range of cosmologies, including cosmologies with

non-negligible neutrino contributions to the total matter density.

Our default analysis assumes that neutrinos have a

negligi-bly small mass. �e only component of our analysis that could

be a�ected by this assumption is when we contrast the

low-redshi� value of σ

derived from clusters with the CMB power

spectrum normalization. �is comparison uses evolution of

purely CDM+baryons power spectra. �e presence of light

neutrinos a�ects the power spectrum at cluster scales; in terms

of σ

, the e�ect is roughly proportional to the total neutrino

density, and is

≈ 20% for ∑ m

= 0.5 eV (we calculate the

ef-fect of neutrinos using the transfer function model of

Eisen-stein & Hu 1999). Stringent upper limits on the neutrino mass

were reported from comparison of the WMAP and Ly-α forest

data, ∑ m

< 0.17 eV at 95% CL (Seljak et al. 2006). If neutrino

masses are indeed this low, they would have no e�ect on our

analysis. However, possible issues with modeling of the Ly-α

data have been noted in the literature (see, e.g., discussion in

§ 4.2.8 of Dunkley et al. 2008) and so we experiment also with

neutrino masses outside the Ly-α forest bounds (§ 8.5).

4.

FITTING PROCEDURE

We obtain parameter constraints using the likelihood

func-tion computed on a full grid of cosmological parameters

a�ect-ing cluster observables (and also those for external datasets).

�e relevant parameters for the cluster data are those that a�ect

the distance-redshi� relation, as well as the growth and power

spectrum of linear density perturbations: Ω

, Ω

, w (dark

en-ergy equation of state parameter), σ

(linear amplitude of

den-sity perturbations at the 8 h

Mpc scale at z

= 0), h, tilt of the

primordial �uctuations power spectrum, and potentially, the

non-zero rest mass of light neutrinos. �is is computationally

demanding and we describe our approach below.

�e computation of the likelihood function for a single

com-bination of parameters is relatively straightforward. Our

pro-cedure (described in Paper II) uses the full information

con-tained in the dataset, without any binning in mass or redshi�,

takes into account the scatter in the M

vs. proxy relations

and measurement errors, and so on. We should note,

how-ever, that since the measurement of the M

and Y

proxies

depends on the assumed distance to the cluster, the mass

func-tions must be re-derived for each new combination of the

cos-mological parameters that a�ect the distance-redshi� relation

— Ω

, w, Ω

, etc. Variations of h lead to trivial rescalings of

the mass function and do not require re-computing the mass

estimates. Computation of the survey volume uses a model for

the evolving L

− M

relation (see § 5 in Paper II), which is

measured internally from the data and thus also depends on

the assumed d

(z) function. �erefore, we re�t the L

− M

relation for each new cosmology and recompute V

(M).

Sen-sitivity of the derived mass function to the background

cos-mology is illustrated in Fig. 2. �e entire procedure, although

equivalent to full reanalysis of the Chandra and ROSAT data,

Figure 1.7: From Vikhlinin et al. (2009), showing the evolution of the mass function givenΛCDM cosmology (solid lines) and the observed cluster abundance distribution (points). Notice the exponential decrement of high mass high redshift clusters. Those cluster are specially sensitive to the cosmology and as such they are excellent probes to recover cosmological parameters.

where P (k) is the linear matter power spectrum, k the wavenumber, and ˆW is the Fourier transform of the real space top-hat window function of radius R. In this way, σM is a function

of a scale or mass given M ∝ R3_{. Notice that with a scale invariant power spectrum P (k) ∝ k,}

σM becomes ∝ k2. This implies that at higher scales (or mass) σM is lower. Due to the effect

of σM in eq. 1.5, the number of high mass clusters is expected to rapidly decrease at higher

and higher masses (see eq. 1.4). This can be seen clearly in Fig. 1.7.

The observable is the average number of clusters of a given mass above a minimum value at a redshift z observed over a dΩ solid angle (Haiman et al., 2001):

dN dzdΩ(z) = dV dzdΩ(z) Z ∞ Mmin(z) dM_dMdn ! (1.7) By comparing eq. 1.4 to observations, through eq. 1.7, cosmological parameters are extracted. In comparison to SN and BAO techniques, which are purely geometrical, Clusters of Galaxies

Figure 1.8: The Bullet Cluster: By using different observational techniques, the different matter components of the galaxy cluster can be seen, the Dark Matter is traced by the weak lensing observation, the baryon gas is traced by the X–rays and the baryons in form of galaxies and stars are seen in optical wavelength. Source: Chandra X-ray observatory site, Harvard University.

13 890 S. W. Allen et al.

Ω

Ω

Figure 6. The 68.3 and 95.4 per cent (1 and 2 σ ) confidence constraints in

the ("m, "#) plane for the Chandra fgasdata (red contours; standard priors

on "bh2and h are used). Also shown are the independent results obtained

from CMB data (blue contours) using a weak, uniform prior on h (0.2 < h < 2), and SNIa data (green contours; the results for the Davis et al. 2007 compilation are shown). The inner, orange contours show the constraint obtained from all three data sets combined (no external priors on "bh2and

hare used). A #CDM model is assumed, with the curvature included as a free parameter.

priors on "bh2 and h, we measure "m= 0.27 ± 0.06 and "#=

0.86 ± 0.19 (68 per cent confidence limits) with χ2

= 41.5 for 40 degrees of freedom. The low χ2_{value obtained is important and}

indicates that the model provides an acceptable description of the data (see Section 5.3 below). The result on "mis in excellent

agree-ment with that determined from the six lowest redshift clusters only (Section 5.1). The result is also consistent with the value reported by Allen et al. (2004) using the previous release of fgasdata, although

the more conservative systematic allowances included here lead to the quoted uncertainties in "mbeing larger by ∼50 per cent.

Fig. 7 shows the marginalized constraints on "#obtained using

both the standard and weak priors on "bh2 and h. We see that

using only the weak priors ("bh2= 0.0214 ± 0.0060, h = 0.72 ±

0.24), the fgasdata provide a clear detection of the effects of dark

energy on the expansion of the Universe, with "#= 0.86 ± 0.21:

a model with "#! 0 is ruled out at ∼99.98 per cent confidence.

(Using the standard priors on "bh2and h, a model with "#! 0 is

ruled out at 99.99 per cent confidence; Table 5.) The significance of the detection of dark energy in the fgasdata is comparable to that

of current SNIa studies (e.g. Riess et al. 2007; Wood-Vasey et al. 2007). The fgasdata provide strong, independent evidence for cosmic

acceleration.

In contrast to the "m constraints, the error budget for "#

in-cludes significant contributions from both statistical and systematic sources. From the analysis of the full sample of 42 clusters using the standard priors on "bh2and h, we find "#= 0.86 ± 0.19; the

error bar comprises ±0.15 statistical error and ±0.12 systematic uncertainty. Thus, whereas improved measurements of "mfrom the

fgas method will require additional information leading to tighter

priors and systematic allowances, significant improvements in the

0 0.5 1 1.5 0 0.5 1 Prob. ΩΛ

Figure 7. The marginalized constraints on "#determined from the

Chan-dra fgasdata using the non-flat #CDM model and standard (solid curve) and

weak (dashed curve) priors on "bh2and h. The fgasdata provide a detection

of the effects of dark energy at the ∼99.99 per cent confidence level.

precision of the dark energy constraints should be possible simply by gathering more data (e.g. doubling the present fgasdata set).

Fig. 6 also shows the constraints on "mand "#obtained from the

CMB (blue contours) and SNIa (green contours) data (Section 4.3). The agreement between the results for the independent data sets is excellent and motivates a combined analysis. The inner, orange contours in Fig. 6 show the constraints on "mand "#obtained from

the combined fgas+ CMB + SNIa data set. We obtain marginalized

68 per cent confidence limits of "m= 0.275 ± 0.033 and "# =

0.735 ± 0.023. Together, the fgas + CMB + SNIa data also

constrain the Universe to be close to geometrically flat: "k =

−0.010 ± 0.011. No external priors on "bh2 and h are used in

the analysis of the combined fgas+ CMB + SNIa data (see also

Section 5.6).

Finally, we have examined the effects of doubling the allowance for non-thermal pressure support in the clusters, i.e. setting 1.0 < γ <1.2. For the analysis of the fgas data alone, this boosts the

best-fitting value of "mby ∼5 per cent but leaves the results on

dark energy unchanged. This can be understood by inspection of equation (3) and recalling that the constraint on "mis determined

primarily from the normalization of the fgascurve, whereas the

con-straints on dark energy are driven by its shape (Section 4.2). For the combined fgas+ CMB + SNIa data set, doubling the width of the

allowance on γ has a negligible impact on the results, since in this case the value of "mis tightly constrained by the combination of

data sets.

5.3 Scatter in the fgasdata

Hydrodynamical simulations suggest that the intrinsic dispersion in

fgasmeasurements for the largest, dynamically relaxed galaxy

clus-ters should be small. Nagai et al. (2007a) simulate and analyse mock X-ray observations of galaxy clusters (including cooling and feed-back processes), employing standard assumptions of spherical sym-metry and hydrostatic equilibrium and identifying relaxed systems based on X-ray morphology in a similar manner to that employed here. For relaxed clusters, these authors find that fgasmeasurements

C

#2007 The Authors. Journal compilation#C2007 RAS, MNRAS 383, 879–896

Figure 1.9: Combining different probes we constrainΩΛ andΩm.

are sensitive to the geometry and the growth of structure.

Each cosmological probe has unique problems and advantages, and characterize from a dif-ferent perspective the Universe. The combination of probes and how they constraint the ΩM− ΩΛ plane can be seen in Fig 1.9.

Organization

This thesis is based on work done as part of the South Pole Telescope collaboration. In Chapter 2 we summarize the main scientific results of this collaboration where the author of this thesis has contributed in different degrees. In Chapter 3 we analyze the third data release, estimate photometric redshifts, identify the Brightest Cluster Galaxies, and compare their distribution to the that from other selection methods such as X-rays. In Chapter 4 we perform a study of the first four blindly SZ-selected galaxy clusters, comparing their optical properties to those of galaxy clusters selected by other means. In Chapter 5 we extend this study of optical properties to a larger sample of the 26 most massive galaxy clusters in the SPT footprint (Williamson et al., 2011) and additionally study the redshift evolution of the galaxy population. Finally, Chapter 6 provides a summary of the results and an outlook for subsequent work for the future.

Chapter

2

Galaxy Clusters, the Sunyaev–Zel’dovich

Effect, and the South Pole Telescope

The thermal Sunyaev–Zel’dovich effect (SZE) was predicted in 1972 in a paper by R.A. Sunyaev and Y.B. Zel’dovich (Sunyaev & Zel’dovich, 1972). The effect is a distortion in the CMB spectrum due to inverse Compton scattering of CMB photons by hot electrons in the intracluster medium. Only about ∼1% of the CMB photons are scattered, so the effect subtle and can be hard to measure. Neglecting relativistic correction, the fractional change in the temperature can be written as

∆TSZE

TCMB = f(x)y = f(x)

Z

nek_mBTe

ec2σTdl, (2.1)

where y, the Compton y-parameter, is a function of the Thomson cross-section σT, the electron

number density ne, the electron temperature Te, the Boltzmann constant kB, and the electron

rest mass energy mec2. The integral is taken along the line of sight. f(x), which is a function

of the reduce frequency (x = hν kBT), is

f(x) = (xe_ex_x+ 1_{− 1}− 4)(1 + δSZE(x, Te)). (2.2)

The intensity of the SZE effect signal is given by ∆ISZE = x

4_ex

ex− 1f(x)I0y. (2.3)

From Eq. 2.1 it can be seen that the distortion is independent of redshift and depends on the value of the integrated pressure of the cluster, which itself depends on the size of the cluster’s potential well, which is determined by the cluster mass. This is the reason why cluster surveys based on the SZE signal are so powerful.

Although a powerful selection method, the SZE is hard to observe. We show the effect of hot cluster electrons to the CMB spectrum in Fig. 2.1. In this Figure we have scaled down the blackbody spectrum of the CMB by a factor of 100 and plotted how the SZE would change it. From this Figure, it is clear that the effect is subtle even if the blackbody spectrum is just 1% of its actual value. Because the change in the spectrum is so small, it was several